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Search: a036722 -id:a036722
Displaying 1-10 of 12 results found. page 1 2
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A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals. +10
15
1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 4, 6, 1, 0, 1, 1, 1, 2, 4, 8, 11, 1, 0, 1, 1, 1, 2, 4, 9, 17, 23, 1, 0, 1, 1, 1, 2, 4, 9, 19, 39, 46, 1, 0, 1, 1, 1, 2, 4, 9, 20, 45, 89, 98, 1, 0, 1, 1, 1, 2, 4, 9, 20, 47, 106, 211, 207, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,19
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
A(n,k) = Sum_{i=0..k} A244372(n,i) for n>0, A(0,k) = 1.
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
0, 1, 3, 4, 4, 4, 4, 4, 4, 4, 4, ...
0, 1, 6, 8, 9, 9, 9, 9, 9, 9, 9, ...
0, 1, 11, 17, 19, 20, 20, 20, 20, 20, 20, ...
0, 1, 23, 39, 45, 47, 48, 48, 48, 48, 48, ...
0, 1, 46, 89, 106, 112, 114, 115, 115, 115, 115, ...
0, 1, 98, 211, 260, 277, 283, 285, 286, 286, 286, ...
0, 1, 207, 507, 643, 693, 710, 716, 718, 719, 719, ...
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
A:= (n, k)-> `if`(n=0, 1, b(n-1$2, k$2)):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[ b[i-1, i-1, k, k]+j-1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]];
A[n_, k_] := If[n == 0, 1, b[n - 1, n - 1, k, k]];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)
PROG
(Python)
from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, i, t, k): return 1 if n==0 else 0 if i<1 else sum([binomial(b(i-1, i-1, k, k)+j-1, j)*b(n-i*j, i-1, t-j, k) for j in range(min(t, n//i)+1)])
def A(n, k): return 1 if n==0 else b(n-1, n-1, k, k)
for d in range(15): print([A(n, d-n) for n in range(d+1)]) # Indranil Ghosh, Mar 02 2018, after Maple code
CROSSREFS
Columns k=1-11 give: A000012, A001190(n+1), A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556.
Main diagonal gives A000081 for n>0.
A(2n,n) gives A299039.
Cf. A244372.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 01 2018
STATUS
approved
A036718 Number of rooted trees where each node has at most 4 children. +10
13
1, 1, 1, 2, 4, 9, 19, 45, 106, 260, 643, 1624, 4138, 10683, 27790, 72917, 192548, 511624, 1366424, 3666930, 9881527, 26730495, 72556208, 197562840, 539479354, 1477016717, 4053631757, 11149957667, 30732671572, 84871652538, 234802661446, 650684226827 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. R. Bremner and H. A. Eigendy, Alternating quaternary algebra structures on irreducible representations of sl_2(C), Lin. Alg. Applic. 433 (2010) 1686-1705.
F. Ruskey, Information on Rooted Trees
FORMULA
G.f. satisfies A(x) = 1 + x*cycle_index(Sym(4), A(x)).
a(n) = Sum_{j=1..4} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017
a(n) / a(n+1) ~ 0.343520104570489046632074698738792654644751898257681287407149... - Robert A. Russell, Feb 11 2023
EXAMPLE
From Joerg Arndt, Feb 25 2017: (Start)
The a(5) = 9 rooted trees with 5 nodes and out-degrees <= 4 are:
: level sequence out-degrees (dots for zeros)
: 1: [ 0 1 2 3 4 ] [ 1 1 1 1 . ]
: O--o--o--o--o
:
: 2: [ 0 1 2 3 3 ] [ 1 1 2 . . ]
: O--o--o--o
: .--o
:
: 3: [ 0 1 2 3 2 ] [ 1 2 1 . . ]
: O--o--o--o
: .--o
:
: 4: [ 0 1 2 3 1 ] [ 2 1 1 . . ]
: O--o--o--o
: .--o
:
: 5: [ 0 1 2 2 2 ] [ 1 3 . . . ]
: O--o--o
: .--o
: .--o
:
: 6: [ 0 1 2 2 1 ] [ 2 2 . . . ]
: O--o--o
: .--o
: .--o
:
: 7: [ 0 1 2 1 2 ] [ 2 1 . 1 . ]
: O--o--o
: .--o--o
:
: 8: [ 0 1 2 1 1 ] [ 3 1 . . . ]
: O--o--o
: .--o
: .--o
:
: 9: [ 0 1 1 1 1 ] [ 4 . . . . ]
: O--o
: .--o
: .--o
: .--o
(End)
MAPLE
A := 1; f := proc(n) global A; local A2, A3, A4; A2 := subs(x=x^2, A); A3 := subs(x=x^3, A); A4 := subs(x=x^4, A);
coeff(series( 1+x*( (A^4+3*A2^2+8*A*A3+6*A^2*A2+6*A4)/2 ), x, n+1), x, n); end;
for n from 1 to 50 do A := series(A+f(n)*x^n, x, n +1); od: A;
MATHEMATICA
a = 1; f[n_] := Module[{a2, a3, a4}, a2 = a /. x -> x^2; a3 = a /. x -> x^3; a4 = a /. x -> x^4; Coefficient[ Series[ 1 + x*(a^4 + 3*a2^2 + 8*a*a3 + 6*a^2*a2 + 6*a4)/24, {x, 0, n + 1}] // Normal, x, n]]; For[n = 1, n <= 30, n++, a = Series[a + f[n]*x^n, {x, 0, n + 1}] // Normal]; CoefficientList[a, x] (* Jean-François Alcover, Jan 16 2013, after Maple *)
b[0, i_, t_, k_] = 1; m = 4; (* m = maximum children *)
b[n_, i_, t_, k_]:= b[n, i, t, k]= If[i<1, 0,
Sum[Binomial[b[i-1, i-1, k, k] + j-1, j]*
b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
PrependTo[Table[b[n-1, n-1, m, m], {n, 1, 30}], 1] (* Robert A. Russell, Dec 27 2022 *)
CROSSREFS
Cf. A000081, A036717, A036719, A036720, A036721, A036722, A182378, A244372.
Cf. A292553, A292554, A292555, A292556.
Column k=4 of A299038.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
Better description from Frank Ruskey, Sep 23 2000
STATUS
approved
A036721 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(5), A(x)). +10
12
1, 1, 1, 2, 4, 9, 20, 47, 112, 277, 693, 1766, 4547, 11852, 31146, 82534, 220149, 590834, 1593951, 4320723, 11761394, 32138301, 88121176, 242383729, 668607115, 1849194691, 5126800907, 14245679652, 39666239726, 110661514973, 309280533011, 865839831118 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Also the number of rooted trees where each node has at most 5 children. [Patrick Devlin, Apr 30 2012]
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Joerg Arndt, The a(6) = 20 rooted trees with 5 nodes and out-degrees <= 5
FORMULA
a(n) = Sum_{j=1..5} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017
a(n) / a(n+1) ~ 0.340017469151060086823930137816585262710976835711484267209811... - Robert A. Russell, Feb 11 2023
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> `if`(n=0, 1, b(n-1$2, 5$2)):
seq(a(n), n=0..35); # Alois P. Heinz, Sep 20 2017
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 5, 5]];
Table[a[n], {n, 0, 35}] // Flatten (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)
CROSSREFS
Cf. A000081, A036717, A036718, A036719, A036720, A036722, A182378, A244372, A292553, A292554, A292555, A292556.
Column k=5 of A299038.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane.
STATUS
approved
A292556 Number of rooted unlabeled trees on n nodes where each node has at most 11 children. +10
12
1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12485, 32970, 87802, 235355, 634771, 1720940, 4688041, 12824394, 35216524, 97039824, 268238379, 743596131, 2066801045, 5758552717, 16080588286, 44997928902, 126160000878, 354349643101, 996946927831 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marko Riedel, Trees with bounded degree.
FORMULA
Functional equation of g.f. is T(z) = z + z*Sum_{q=1..11} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group.
Alternate FEQ is T(z) = 1 + z*Z(S_11)(T(z)).
a(n) = Sum_{j=1..11} A244372(n,j) for n > 0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
Limit_{n->oo} a(n)/a(n+1) = 0.338324339068091181557475416836618315086769320447748735003402... - Robert A. Russell, Feb 11 2023
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> `if`(n=0, 1, b(n-1$2, 11$2)):
seq(a(n), n=0..35); # Alois P. Heinz, Sep 20 2017
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[ b[i-1, i-1, k, k]+j-1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n-1, n-1, 11, 11]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)
CROSSREFS
Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292553, A292554, A292555.
Column k=11 of A299038.
KEYWORD
nonn
AUTHOR
Marko Riedel, Sep 18 2017
STATUS
approved
A292553 Number of rooted unlabeled trees on n nodes where each node has at most 8 children. +10
11
1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 718, 1839, 4757, 12460, 32897, 87592, 234746, 633013, 1715851, 4673320, 12781759, 35093010, 96681705, 267199518, 740580555, 2058042803, 5733101603, 16006590851, 44782679547, 125533577578, 352525803976, 991634575368 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marko Riedel, Trees with bounded degree.
Marko Riedel, Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 1).
Marko Riedel, Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 2)
Marko Riedel, Maple code (FEQ 2) optimized for speed.
FORMULA
Functional equation of G.f. is T(z) = z + z*Sum_{q=1..8} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is T(z) = 1 + z*Z(S_8)(T(z)).
a(n) = Sum_{j=1..8} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
a(n) / a(n+1) ~ 0.338386042364849957035744926227166370702775721795018600630554... - Robert A. Russell, Feb 11 2023
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> `if`(n=0, 1, b(n-1$2, 8$2)):
seq(a(n), n=0..35); # Alois P. Heinz, Sep 20 2017
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 8, 8]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)
CROSSREFS
Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292554, A292555, A292556.
Column k=8 of A299038.
KEYWORD
nonn
AUTHOR
Marko Riedel, Sep 18 2017
STATUS
approved
A292554 Number of rooted unlabeled trees on n nodes where each node has at most 9 children. +10
11
1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1841, 4763, 12477, 32947, 87735, 235162, 634212, 1719325, 4683368, 12810871, 35177357, 96926335, 267909285, 742641309, 2064029034, 5750500663, 16057186086, 44929879114, 125962026154, 353773417487, 995269027339 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marko Riedel, Trees with bounded degree.
FORMULA
Functional equation of G.f. is T(z) = z + z*Sum_{q=1..9} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is
T(z) = 1 + z*Z(S_9)(T(z)).
a(n) = Sum_{j=1..9} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
a(n) / a(n+1) ~ 0.338343552789108712866488147828528012266693326385052387884853... - Robert A. Russell, Feb 11 2023
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> `if`(n=0, 1, b(n-1$2, 9$2)):
seq(a(n), n=0..35); # Alois P. Heinz, Sep 20 2017
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 9, 9]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)
CROSSREFS
Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292553, A292555, A292556.
Column k=9 of A299038.
KEYWORD
nonn
AUTHOR
Marko Riedel, Sep 18 2017
STATUS
approved
A292555 Number of rooted unlabeled trees on n nodes where each node has at most 10 children. +10
11
1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4765, 12483, 32964, 87785, 235305, 634628, 1720524, 4686842, 12820920, 35206475, 97010705, 268154003, 743351390, 2066090876, 5756490561, 16074597300, 44980514021, 126109353817, 354202275766, 996517941454 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marko Riedel, Trees with bounded degree.
FORMULA
Functional equation of G.f. is T(z) = z + z*Sum_{q=1..10} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is
T(z) = 1 + z*Z(S_10)(T(z)).
a(n) = Sum_{j=1..10} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
a(n) / a(n+1) ~ 0.338329194566131211670667671160855741193081902868090986608524... - Robert A. Russell, Feb 11 2023
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> `if`(n=0, 1, b(n-1$2, 10$2)):
seq(a(n), n=0..35); # Alois P. Heinz, Sep 20 2017
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 10, 10]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)
CROSSREFS
Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292553, A292554, A292556.
Column k=10 of A299038.
KEYWORD
nonn
AUTHOR
Marko Riedel, Sep 18 2017
STATUS
approved
A182378 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(7), A(x)). +10
10
1, 1, 1, 2, 4, 9, 20, 48, 115, 285, 716, 1833, 4740, 12410, 32754, 87176, 233547, 629540, 1705809, 4644231, 12697500, 34848694, 95973026, 265142431, 734606478, 2040683413, 5682634446, 15859800889, 44355531103, 124290064228, 348904212741, 981082979409 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Number of rooted trees where each node has at most 7 children.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{j=1..7} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017
a(n) / a(n+1) ~ 0.338512011286603947719604869750539045616436718225097926729820... - Robert A. Russell, Feb 11 2023
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> `if`(n=0, 1, b(n-1$2, 7$2)):
seq(a(n), n=0..35); # Alois P. Heinz, Sep 20 2017
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[ b[i-1, i-1, k, k] + j - 1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n-1, n-1, 7, 7]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 15 2018, after Alois P. Heinz *)
CROSSREFS
Cf. A000081, A036717, A036718, A036719, A036720, A036721, A036722, A244372, A292553, A292554, A292555, A292556.
Column k=7 of A299038.
KEYWORD
nonn
AUTHOR
Michael Burkhart, Apr 26 2012
EXTENSIONS
More terms from Patrick Devlin, Apr 29 2012
STATUS
approved
A244402 Number of unlabeled rooted trees with n nodes and maximal outdegree (branching factor) 6. +10
2
1, 2, 6, 17, 50, 143, 415, 1192, 3444, 9931, 28687, 82857, 239563, 692878, 2005381, 5806915, 16824277, 48767953, 141430699, 410341703, 1191064873, 3458607705, 10046993035, 29196507434, 84874753458, 246814998803, 717965190047, 2089140528083, 6080768466919 (list; graph; refs; listen; history; text; internal format)
OFFSET
7,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 7..1000
FORMULA
a(n) = A036722(n) - A036721(n).
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> b(n-1$2, 6$2) -`if`(k=0, 0, b(n-1$2, 5$2)):
seq(a(n), n=7..40);
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]* b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]] // FullSimplify] ; a[n_] := b[n - 1, n - 1, 6, 6] - If[n == 0, 0, b[n - 1, n - 1, 5, 5]]; Table[a[n], {n, 7, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)
CROSSREFS
Column k=6 of A244372.
Cf. A036721, A036722.
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 27 2014
STATUS
approved
A244403 Number of unlabeled rooted trees with n nodes and maximal outdegree (branching factor) 7. +10
2
1, 2, 6, 17, 50, 143, 416, 1198, 3467, 10019, 29001, 83945, 243228, 705012, 2044935, 5934425, 17231410, 50058023, 145491836, 423056364, 1230683672, 3581556220, 10427172296, 30368394833, 88476965536, 257860132679, 751756288476, 2192311994070, 6395199688864 (list; graph; refs; listen; history; text; internal format)
OFFSET
8,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 8..1000
FORMULA
a(n) = A182378(n) - A036722(n).
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> b(n-1$2, 7$2) -`if`(k=0, 0, b(n-1$2, 6$2)):
seq(a(n), n=8..40);
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]* b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]] // FullSimplify] ; a[n_] := b[n - 1, n - 1, 7, 7] - If[n == 0, 0, b[n - 1, n - 1, 6, 6]]; Table[a[n], {n, 8, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)
CROSSREFS
Column k=7 of A244372.
Cf. A036722, A182378.
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 27 2014
STATUS
approved
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Last modified August 6 16:57 EDT 2024. Contains 374980 sequences. (Running on oeis4.)